It is commonly believed that the energy level spacings of integrable systems follow a Poisson distribution, while those of classically chaotic systems follow Wigner-Dyson statistics instead. Someone mentioned to me recently that actually that although the presence of Poisson statistics is often used as a "diagnostic" for integrability, it is actually a necessary but not sufficient condition. Unfortunately I was not able to question them further to get more details.
So my question is: Is it true that Poissonian level statistics are just a necessary, but not sufficient, condition for integrability? As a corollary, can you give me an example of a system that has Poisson statistics, but is non-integrable?
